I am looking for a while to prove or disprove: (preparing for finals) O(f(n)-g(n)) ⊂ |O(f(n)) - O(g(n))| where || is absolute value. Note that ⊂ is needed and not ⊆
I assumed the a subtraction operator between 2 O() means:
t1(n) is O(f(n)) and t2(n) is O(g(n)) and so t1(n)-t2(n)≤O(f(n)) - O(g(n))
I also noted that f(n)≥g(n) otherwise the left side of the proof is invalid.
I am n ot sure how to proceed.
I started at the left side with:
t(n) is O(f(n)-g(n))
t(n)≤c*(f(n)-g(n)) (c>0,n≥n0)
t(n)≤c*f(n)-c*g(n)
I dont know how to proceed from here and how to prove that the right side has a member that doesnt belong to the left side (proper subset rule) Thank you for any assistance.